Does the men's rights movement engage in identity politics?

No. One of the greatest misunderstandings about the men's rights movement and anti-feminism is that they in any way involve identity politics. Merely seeking to address the problems faced by a particular group does not imply identity politics.

The Oxford Dictionary defines identity politics as:

A tendency for people of a particular religion, race, social background, etc., to form exclusive political alliances, moving away from traditional broad-based party politics.

As can be seen from this, and other definitions, identity politics is inherently exclusionary. The men's rights movement accepts anyone who wants to help us regardless of their gender, ethnicity, sexuality or any other characteristic they might have. Every major men's rights organisation has women in leadership roles. This is because those women were willing and able to fill those positions. Anyone who wants to help MRAs address the problems facing men and boys is welcome.

Compare to feminist groups. Many exclude men from leadership roles and some exclude men entirely - calling them 'allies'. Some don't even want male allies. That's identity politics.